Abstract :
Starting from previous results concerning determinants and permanents of (0,1) circulant matrices, and using theory on finite fields with characteristic 2, we first see (denoting by A the n×n (0,1) circulant matrix whose 1’s in the first row are exactly in positions i1,i2,…,ik) that Per(A) is even iff gcd(xi1+xi2+cdots, three dots, centered+xik,1+xn)≠1 in Z2[x].
We then derive some consequences for specific classes of primes n: in particular, when n is a prime such that the group image is generated by the residue class 2, we see that for k≠n and k odd the value Per(A) is always odd. Similarly, for n>7 and n prime with the group of squares in image generated by the class 2, we see that for k=3 the value Per(A) is always odd. Contrary to such cases, we find that when n is a Mersenne prime greater than 3, there are a significant number of circulants A of the above form with k odd and Per(A) even.
